During TNT's broadcast of the 2022 Winter Classic between the St. Louis Blues and Minnesota Wild, NHL commissioner Gary Bettman stated that he's explored the idea of having hockey added to the Summer Olympics.
According to Bettman, he's been trying to make this happen since the mid-1990s, but the International Olympic Committee has been adamant about not moving it. Hockey made its Olympic debut at the 1920 Summer Games before switching to the Winter Olympics in 1924.
A move to the Summer Olympics would obviously make more logistical sense for the NHL, as the Games wouldn't interfere with the league's 82-game regular season or playoffs.